The reported attack came despite Russia agreeing to a 12-hour pause in hostilities to allow refugees to evacuate a number of towns and cities.
Citizens trying to escape shelling on the outskirts of Kyiv streamed toward the capital amid warnings from the West that Moscow's invasion is about to take a more brutal and indiscriminate turn.
READ MORE: Chernobyl site knocked off power grid
Video shared by Zelenskyy showed cheerfully painted hallways strewn with twisted metal and room after room with blown-out windows. Floors were covered in wreckage. Outside, a small fire burned, and debris covered the ground.
Mariupol's city council said on its social media site that the damage was "colossal".
The council posted a video of the devastated hospital in the city and accused Russian forces of dropping several bombs on it from the air.
"The destruction is enormous," said the council. "The building of the medical facility where the children were treated recently is completely destroyed."
Zelenskyy repeated his call for the NATO military alliance to declare a no-fly zone over Ukraine.
"Direct strike of Russian troops at the maternity hospital," he said on Telegram. "People, children are under the wreckage. Atrocity! How much longer will the world be an accomplice ignoring terror?
"Close the sky right now! Stop the killings! You have power but you seem to be losing humanity."
Authorities said they were trying to establish how many people had been killed or wounded.
The strategic port city of Mariupol on Ukraine's southeast coast has been under siege for days and has been "isolated" by Russian forces, a senior US defence official said Tuesday.
Authorities announced new cease-fires Wednesday morning to allow thousands of civilians to escape from towns around Kyiv as well as the southern cities of Mariupol, Enerhodar and Volnovakha, Izyum in the east and Sumy in the northeast.
Previous attempts to establish safe evacuation corridors largely failed because of what the Ukrainians said were Russian attacks. But Russian President Vladimir Putin, in a telephone call with German's chancellor, accused militant Ukrainian nationalists of hampering the evacuations.
It was not immediately clear whether anyone was able to leave other cities on Wednesday, but people streamed out of Kyiv's suburbs, many headed for the city centre, even as explosions were heard in the capital and air raid sirens sounded repeatedly. From there, the evacuees planned to board trains bound for western Ukrainian regions not under attack.
Civilians trying to escape the Kyiv suburb of Irpin were forced to make their way across the slippery wooden planks of a makeshift bridge, because the Ukrainians blew up the concrete span to Kyiv days ago to slow the Russian advance,
With sporadic gunfire echoing behind them, firefighters dragged an elderly man to safety in a wheelbarrow, a child gripped the hand of a helping soldier, and a woman inched her way along cradling a fluffy cat inside her winter coat.
They trudged past a crashed van with the words "Our Ukraine" written in the dust coating its windows.
"We have a short window of time at the moment,'' said Yevhen Nyshchuk, a member of Ukraine's territorial defence forces.
"Even if there is a cease-fire right now, there is a high risk of shells falling at any moment."
In Mariupol, local authorities hurried to bury the dead in a mass grave. City workers dug a trench some 25 metres long at one of the city's old cemeteries and made the sign of the cross as they pushed bodies wrapped in carpets or bags over the edge.
Nationwide, thousands are thought to have been killed, both civilians and soldiers, in the two weeks of fighting since Putin's forces invaded.
The UN estimates more than 2 million people have fled the country, the biggest exodus of refugees in Europe since the end of World War II.
The fighting knocked out power to the decommissioned Chernobyl nuclear plant, raising fears about the spent fuel that is stored at the site and must be kept cool. But the U.N. nuclear watchdog agency said it saw "no critical impact on safety" from the loss of power.
The crisis in Ukraine is likely to get worse as Russian forces step up their bombardment of cities in response to stronger than expected resistance.
Russian losses have been "far in excess" of what Putin and his generals expected, CIA Director William Burns said Tuesday.
An intensified push by Russian forces could mean "an ugly next few weeks," Burns told a congressional committee, warning that Putin is likely to "grind down the Ukrainian military with no regard for civilian casualties."
British Defence Secretary Ben Wallace echoed those comments Wednesday, telling British lawmakers that Russia's assault will get "more brutal and more indiscriminate" as Putin tries to regain momentum.
Britain's Defence Ministry said fighting continued northwest of Kyiv. The cities of Kharkiv, Chernihiv, Sumy and Mariupol were being heavily shelled and remained encircled by Russian forces.
Russian forces are placing military equipment on farms and amid residential buildings in the northern city of Chernihiv, Ukraine's military said. In the south, Russians in civilian clothes are advancing on the city of Mykolaiv, a Black Sea shipbuilding centre of a half-million people, it said.
The Ukrainian military, meanwhile, is building up defences in cities in the north, south and east, and forces around Kyiv are "holding the line" against the Russian offensive, authorities said.
In Irpin, a town of 60,000, police officers and soldiers helped elderly residents from their homes. One man was hoisted out of a damaged structure on a makeshift stretcher, while another was pushed toward Kyiv in a shopping cart. Fleeing residents said they had been without power and water for the past four days.
Regional administration head Oleksiy Kuleba said the crisis for civilians is deepening in and around Kyiv, with the situation particularly dire in the suburbs.
"Russia is artificially creating a humanitarian crisis in the Kyiv region, frustrating the evacuation of people and continuing shelling and bombing small communities," he said.
The situation is even worse in Mariupol, a strategic city of 430,000 people on the Sea of Azov that has been encircled by Russian forces for the past week.
Efforts to evacuate residents and deliver badly needed food, water and medicine failed Tuesday because of what the Ukrainians said were continued Russian attacks.
The city took advantage of a lull in the shelling Wednesday to hurriedly bury 70 people. Some were soldiers, but most were civilians.
The work was conducted efficiently and without ceremony. No mourners were present, no families to say their goodbyes.
One woman stood at the gates of the cemetery to ask whether her mother was among those being buried. She was.